Volume 1, Issue 2

In this paper we exploit some compactness properties of the set of quasiminimizers of a functional with free discontinuities with the aim of obtaining blow-up techniques. These techniques are finally applied to show regularity near the boundary of the minimizers of the Mumford-Shah functional.

We establish the existence of solutions of the problem under various assumptions on the level sets of Q(x), where Ω is a smooth bounded domain in ℝ N , N ≥ 3, Q(x) 0, and a(x) are smooth functions. We also show that if a(x) ≥ a 0 > 0 and a 0 is suiliciently large, our solutions are not ground states.

We study the notion of distributional chaos for a class of two-dimensional maps called Cournot maps, which models a competitive economic process (Cournot duopoly). Among other results, we prove that a Cournot map has positive topological entropy if and only if it is distributionally chaotic.

We study the solvability and the existence of multiple solutions of nonlinear systems of elliptic equations.

Let Ω be an open bounded subset of ℝ N and b a measurable nonnegative function in ℝ. We study the time compact support property for ∂ t u - Δ p u + b(x)u q = 0 with p > 2 and for ∂ t u - Δ(u m ) + b(x)u q = 0 with m > 1 where 0 ≤ q < 1. Some criteria involving the asymptotics as h → 0 of the first eigenvalue of are given which imply such a phenomenon. Applying these criteria we deduce that the time vanishing property occurs when 1/b belongs to L s (Ω) for some s depending on q, and p or m.